Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 3 / a

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- Solution a

Solution

For a continuous local martingale

M

with

M_{0} = 0

, its stochastic exponential is

E (M)_{t} = exp (M_{t} - \frac{1}{2} [M]_{t}) .

The Itô formula gives

d E (M)_{t} = E (M)_{t} d M_{t}

, so

E (M)

is a positive continuous local martingale. Every nonnegative local martingale is a supermartingale, because localization and the Conditional Fatou lemma turn the localized martingale equality into the supermartingale inequality.

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